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A natural transformation in a set-valued functor category is an epimorphism exactly when every component is a surjective function. More generally, limits and colimits in a functor category are computed pointwise whenever they exist in the codomain.

An object $P$ is projective when every morphism $P\to B$ lifts through every epimorphism $A\twoheadrightarrow B$. Equivalently, the functor $\mathcal{C}(P,-)$ preserves epimorphisms.

A morphism $e:A\to B$ in a pointed category is pseudo-epic when, for every $t:B\to Z$, the equation $te=0$ forces $t=0$. If a categorical cokernel exists, this is equivalent to that cokernel having a zero object as codomain. Indeed all zero composites then factor uniquely through zero; conversely the cokernel itself must be zero, and being both zero and epic forces its codomain to have identity zero, hence be a zero object. Every epimorphism is pseudo-epic. In a preadditive category, subtraction shows the converse: $ue=ve$ implies $(u-v)e=0$, hence $u=v$. In a general pointed category no subtraction is available.

In an abelian category, suppose a commutative square $gf=kh$ is a pullback in a category and $g$ is an epimorphism. Then $(f,h):A\to B\oplus C$ is the categorical kernel of the epimorphism $[g,-k]$. Since an epimorphism is the cokernel of its kernel, any compatible pair $u:B\to X$, $v:C\to X$ induces a unique map $D\to X$ by factoring $[u,-v]$. Hence the square is also a pushout in a category.

In an abelian category, the pullback in a category of an epimorphism $p:B\to C$ along $k:C^{\prime }\to C$ is the kernel of the epimorphism $[p,-k]:B\oplus C^{\prime }\to C$. If $v:C^{\prime }\to D$ annihilates its projection, $[0,v]$ annihilates that kernel and therefore factors through $[p,-k]$. Restriction to $B$ forces the factor to vanish, giving $v=0$ and proving the projection epic. This uses the definition's normality of epimorphisms.

For a natural transformation $a:X\to Y$ of set-valued functors, form the pushout by taking two copies of each $Y(C)$ and identifying the two copies of every element in the image of $a_C$. Naturality makes all induced maps well-defined. If $y\in Y(C)$ is outside that image, its two copies remain distinct. Thus the two maps $Y\rightrightarrows Y{⨿}_{X}Y$ agree after $a$ but differ at $y$, proving $a$ is not an epimorphism. Conversely, componentwise surjective functions allow cancellation at every object, proving that $a$ is an epimorphism. This argument does not assume a natural choice of preimages.

In the covariant functor category $[\mathbb{N},\mathbf{Set}]$, the classifier at $n$ consists of thresholds $k\ge n$ and the empty threshold $\infty$. Its transition to $n+1$ sends $k$ to $\max (k,n+1)$. The displayed maps commute with transitions and are all surjective, so define an epimorphism. They identify $n$ and $n+1$, so the endomorphism is not an isomorphism.

A short exact sequence in an abelian category has one nonzero morphism entering the middle object and one leaving it: $0\to A\stackrel{i}{\to }B\stackrel{p}{\to }C\to 0$. Equivalently, $i$ is the kernel in a category of the epimorphism $p$. It splits if $p$ has a section; in that case the additive category splitting argument identifies $B$ with the biproduct $A\oplus C$. Categories of modules give familiar examples.

A morphism $m:A\to B$ is strict when every $g:C\to B$ satisfying $hm=km\Rightarrow hg=kg$ for all parallel arrows $h,k$ out of $B$ factors uniquely through $m$. Every regular monomorphism is strict by its equalizer property. A strict monomorphism is strong: in a lifting square with an epimorphism, cancellation of the epimorphism makes the bottom arrow admissible, and its unique factorization gives the diagonal.

A strong monomorphism has the right lifting property with respect to every epimorphism. In every commutative square with an epimorphism on the left and the strong monomorphism on the right, there is a unique diagonal filler; uniqueness follows from monicity.

A category is well-copowered if, for each object $C$, the epimorphisms $C\to Q$ have only a set of equivalence classes, where two are equivalent through an isomorphism of their codomains compatible with the quotient maps. Equivalently, its opposite category is well-powered.

In an abelian category, a morphism is an epimorphism if and only if its categorical cokernel object is zero. For the reverse implication, any difference of maps annihilating the morphism factors through its cokernel and is therefore zero. The cokernel invariance under pushout then proves both preservation and reflection of epimorphisms by pushout.