 Solution

Let $T:{\mathcal{C}}^{\mathrm{op}}\times \mathcal{C}\to \mathcal{D}$. A categorical end is an object $E$ with morphisms $e_A:E\to T(A,A)$ satisfying, for every $f:A\to B$,It is universal among such families: any $x_A:X\to T(A,A)$ satisfying the same equations is uniquely $e_{A}h$ for one $h:X\to E$. This is a universal dinatural transformation from the constant value $E$, and is written $E={\int }_{A}T(A,A)$.

Dually a categorical coend is $Q$ with morphisms $j_A:T(A,A)\to Q$ satisfyingAny family $x_A:T(A,A)\to X$ satisfying these equations is uniquely $h{j}_A$ for one $h:Q\to X$. Write $Q={\int }^{A}T(A,A)$. For small $\mathcal{C}$, the categorical end can be constructed as the equalizer of ${\prod }_{A}T(A,A)\rightrightarrows {\prod }_{f:A\to B}T(A,B)$ when these objects exist. The categorical coend is the coequalizer of ${\coprod }_{f:A\to B}T(B,A)\rightrightarrows {\coprod }_{A}T(A,A)$ when those objects exist. The two morphisms impose exactly the displayed equations for a dinatural transformation.