Existence of a Coend in a Monoidal Category

Question

Let $B$ be a monoidal category with multiplication $\Box$. Let $P$ be a category and let $T \colon P^\mathrm{op} \to B$ and $S \colon P \to B$ be functors. MacLane [CWM, p226] says that these two functors have a "tensor product"

$$ T \Box_P S = \int^{p\colon P} (Tp) \Box (Sp) .$$

Is that coend guaranteed to exist? Do we need more assumptions on the structure of $B?$

Answer 1

I guess that (wisely, otherwise any mathematical text would become unreadable) Mac Lane decided not to put all the more or less obvious hypotheses every time they're needed. In this case, since coends are colimits the hypothesis on $B$ would be to be cocomplete.