The hom-functor turns a coend into an end and carries an end to an end

Given: A small $\mathcal{C}$, a locally small $\mathcal{D}$, a functor $T$ on ${\mathcal{C}}^{\mathrm{o}\mathrm{p}}\times \mathcal{C}$ with values in $\mathcal{D}$, and an object $X$ of $\mathcal{D}$.

A category is small when both $\mathrm{Ob}(\mathcal{C})$ and $\mathrm{Mor}(\mathcal{C})$ are sets. (Small, locally small, and large categories).

The objects of $\mathrm{Tw}(\mathcal{C})$ are the morphisms of $\mathcal{C}$, and a morphism $f\to g$ is a pair $(a,b)$ of morphisms of $\mathcal{C}$ with $bfa=g$ (The twisted arrow category and its projection to ${\mathcal{C}}^{\mathrm{o}\mathrm{p}}\times \mathcal{C}$).

The opposite category has the same objects and reverses every morphism, ${\mathcal{C}}^{\mathrm{o}\mathrm{p}}(A,B)=\mathcal{C}(B,A)$, and $({\mathcal{C}}^{\mathrm{o}\mathrm{p}}{)}^{\mathrm{o}\mathrm{p}}=\mathcal{C}$ strictly (Opposite category ${\mathcal{C}}^{\mathrm{o}\mathrm{p}}$).

The hom-assignment sends $(a,b)$ to $\mathcal{C}(a,b)$, and a morphism of the product category consisting of $h:a^{\mathrm{\prime }}\to a$ and $u:b\to b^{\mathrm{\prime }}$ acts by $\mathcal{C}(h,u):\mathcal{C}(a,b)⟶\mathcal{C}(a^{\mathrm{\prime }},b^{\mathrm{\prime }}),f⟼u\circ f\circ h$ (The covariant and contravariant hom-assignments and the hom-bifunctor of a locally small category).

The wedges over a functor are exactly the cones over its composite with the twisted arrow projection, and the cowedges are the cocones under the composite with the swapped projection, so an end is the limit over the twisted arrow category and a coend a colimit over its opposite (An end is a limit over the twisted arrow category, and a coend is a colimit over its opposite).

For every object $X$ of a locally small category and every small diagram $D$ with a colimit there is a natural bijection $\mathcal{C}(\mathrm{colim}D,X)\cong {\lim }_{j\in {\mathcal{J}}^{\mathrm{o}\mathrm{p}}}\mathcal{C}(D(j),X)\mathrm{.}$ (Hom(X,−) is continuous, while Hom(−,X) sends every existing small colimit to a limit of sets).

For every object $X$ of a locally small category $\mathcal{C}$, the covariant hom-functor $\mathcal{C}(X,-)$ preserves all small limits that exist (Hom(X,−) is continuous, while Hom(−,X) sends every existing small colimit to a limit of sets).

If $F$ preserves $\mathrm{Tw}(\mathcal{C})$-limits and $T$ has an end, then $F$ of that end is an end of $FT$, so a functor preserving twisted-arrow limits preserves ends (A functor preserving twisted-arrow limits preserves ends, and dually for coends).

An end of $T$ is a terminal object of the category of wedges over $T$ and a coend an initial object of the category of cowedges under $T$; in short, an end is a terminal wedge and a coend an initial cowedge (The end and the coend of a functor ${\mathcal{C}}^{\mathrm{o}\mathrm{p}}\times \mathcal{C}\to \mathcal{D}$).