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The colimit of two arrows $A\to B$ and $A\to C$. A map from the pushout is uniquely a pair of maps from $B$ and $C$ agreeing on $A$. In modules it is $(B\oplus C)/\{(f(a),-g(a)):a\in A\}$.

A pushout of $B\leftarrow A\to C$ is a colimit with morphisms $B\to P$ and $C\to P$ making the square commute and universal among such commuting pairs. In the Category of sets, form the disjoint union of $B,C$ and identify the two images of each element of $A$.

A functor $F:I\to J$ is initial if restriction preserves categorical limits for diagrams in a category with codomain ${\mathbf{Set}}^{\mathrm{op}}$. To test necessity, view $J(-,j)$ as such a diagram. Its limit is a singleton, since the colimit of the corresponding representable presheaf is the connected-component set of the slice $J\downarrow j$, which has a terminal object. The restricted colimit is the connected-component set of $(F\downarrow j)$. Thus that comma category is nonempty and connected. Sufficiency follows from cone restriction along an initial functor.

A functor $F:\mathcal{X}\to \mathcal{A}$ preserving the possibly large colimits of the projections $U_A:(F\downarrow A)\to \mathcal{X}$ has a right adjoint exactly when all those colimits exist. The universal cocone makes its vertex a terminal object of $(F\downarrow A)$, hence a representation of $\mathcal{A}(F-,A)$.

An additive functor between abelian categories is right exact functor when it sends every exact sequence $X\to Y\to Z\to 0$ to an exact sequence $FX\to FY\to FZ\to 0$. Equivalently, it preserves cokernels in a category, since additivity already preserves finite biproducts. This is the convention used in constructing left derived functors. Left exactness preserves the initial part $0\to X\to Y\to Z$ instead. Contravariant Hom becomes a right exact functor covariant functor with target the opposite abelian category.

An additive functor between abelian categories is right exact when it preserves categorical cokernels, equivalently when it sends every exact sequence ending in zero to an exact sequence ending in zero. Additivity and preservation of cokernels in a category imply preservation of finite colimits. The additivity assumption is part of the ordinary left derived functor construction.

Let $\mathcal{C}$ be the ordered category of all ordinals with an added greatest object $\infty$. The categorical presheaf with $X(\alpha )=\{\ast \}$ at ordinals and $X(\infty )=\varnothing$ preserves every small categorical limit: a small supremum is an ordinal unless its diagram contains $\infty$. Its category of elements consists of all ordinals, and their projection has the large colimit $\infty$. Nevertheless $X$ is not representable: an ordinal representer has bounded support, while the representer $\infty$ has a nonempty value at $\infty$. Thus preservation must include the possibly large opposite elements diagram. The dual functor $X^{\mathrm{op}}:\mathcal{C}\to {\mathbf{Set}}^{\mathrm{op}}$ also disproves the comma-projection criterion for a right adjoint under small-colimit preservation alone.

An object with the displayed property is called small projective; the term 0-presentable here uses this all-small-colimits convention. In a presheaf category, every representable presheaf has this property by the Yoneda lemma and pointwise computation of colimits. Conversely the canonical colimit presentation of a presheaf shows that the identity on such an object factors through one representable, making it a retract. If the indexing category has equalizers, a representable retract is again representable by the representable retract criterion.