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For parallel arrows $f,g:B\rightrightarrows C$, a split coequalizer consists of $q:C\to Q$ and arrows $s:Q\to C$, $t:C\to B$ with $qf=qg$, $qs=1_Q$, $ft=1_C$, and $gt=sq$. These equations prove the coequalizer property: if $uf=ug$, then $u=uft=ugt=usq$, so $us$ is the unique factor through $q$. Every functor preserves this split coequalizer because it preserves these equations.

A split epimorphism $r:B\to A$ has a right inverse $s:A\to B$ with $rs=1_A$. Every functor preserves split epimorphisms.

A split equalizer consists of $e:E\to X$ and $f,g:X\rightrightarrows Y$ with $fe=ge$, together with maps $r:X\to E$, $s:Y\to X$ satisfying $re=1_E$, $sf=er$, and $sg=1_X$. If $fh=gh$, these identities give $h=erh$, so $rh$ is its unique factorization through $e$. Thus it is an equalizer, and every functor preserves it: all the defining identities survive application of the functor. Interchanging $f,g$ gives the other orientation of the same convention.

A split monomorphism $f:A\to B$ has a left inverse $r:B\to A$ with $rf=1_A$. Every functor preserves split monomorphisms.

In a sheaf topos on a topological space, subterminal sheaves correspond to open sets $U$. Their sections over $V$ are a singleton if $V\subseteq U$ and empty otherwise. An open cover gives a jointly epimorphic family of these subobjects. The open cover criterion for a local sheaf topos follows by applying a colimit-preserving global-sections functor to that family.

The category with one object and only its identity morphism. A functor from it selects an object of the target category.

The free-forgetful adjunction for the Eilenberg-Moore category is terminal in the category of adjunctions inducing a fixed monad. The unique morphism into it is the Eilenberg-Moore comparison functor $B\mapsto (GB,G{\epsilon }_B)$. Its underlying objects and arrows are forced by the forgetful functor, and counit compatibility forces its algebra actions.

For a contravariant pointed functor on based CW homotopy types, half-exactness on a cofibration means that $F(X/A)\to F(X)\to F(A)$ is exact at $F(X)$: the image is the inverse image of the distinguished point. This does not demand injectivity or surjectivity at the ends. Set-valued exactness and group-valued exactness must be distinguished.

Two factorizations of one functor have a unique connecting functor in each direction, by the orthogonality of final functors and discrete fibrations. Their composites and the identity fill the same lifting square, so uniqueness makes both composites identities. The intermediate categories are therefore canonically isomorphic.