= 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 $f:A\to B$,
$$
T(1_A,f)e_A=T(f,1_B)e_B.
$$
It is universal among such families: any $x_A:X\to T(A,A)$ satisfying the same equations is uniquely $e_Ah$ 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> $j_A:T(A,A)\to Q$ satisfying
$$
j_BT(1_B,f)=j_AT(f,1_A):T(B,A)\to Q.
$$
Any 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 $\prod_AT(A,A)\rightrightarrows\prod_{f:A\to B}T(A,B)$ when these objects exist. The <categorical coend> is the <coequalizer> of $\coprod_{f:A\to B}T(B,A)\rightrightarrows\coprod_AT(A,A)$ when those objects exist. The two <morphisms> impose exactly the displayed equations for a <dinatural transformation>.
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