= Solution
Let $T:\mathcal C^{\mathrm{op}}\times\mathcal C\to\mathcal D$. An <end of a functor> is an object $E$ with maps $e_C:E\to T(C,C)$ satisfying $T(1_C,f)e_C=T(f,1_D)e_D$ for every $f:C\to D$, universal among all such families. Thus every compatible family with vertex $A$ factors uniquely through $E$. Write it as $\int_CT(C,C)$. These are the wedge equations for a <dinatural transformation> from a constant <functor> to $T$.
Dually, a <coend of a functor> is an object $Q$ with maps $i_C:T(C,C)\to Q$ satisfying $i_CT(f,1_C)=i_DT(1_D,f)$ as maps from $T(D,C)$, universal among compatible families with a common target. Write it as $\int^CT(C,C)$. These are cowedge equations, and every cowedge to $A$ factors uniquely through $Q$.
For small $\mathcal C$, if the needed products and <equalizers> exist, the end is the <equalizer> of $\prod_CT(C,C)\rightrightarrows\prod_{f:C\to D}T(C,D)$. If the needed coproducts and <coequalizers> exist, the coend is the <coequalizer> of $\coprod_{f:C\to D}T(D,C)\rightrightarrows\coprod_CT(C,C)$. The two arrows in each construction are precisely the two sides of the compatibility equation. This explains both the mixed variance and the <universal property>, rather than treating the integral notation as an ordinary numerical integral.
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