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I asked a very similar question but it was suggested that I rephrase and re-ask it.

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"

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

It is not obvious to me that this coend exists. What are the correct assumptions for this to be true? Should it be

If $B$ is cocomplete and $P$ is small then...

or maybe

If $B$ is cocomplete and $P$ is any category...

I am very interested in the second case and I was wondering if this is known to be true or not. Note that it does not make sense to assume $B$ has all (co)limits since necessarily this would force $B$ to be a preorder. So cocomplete of course means "has all small colimits".

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    $B$ cocomplete and $P$ small is a sufficient condition. $P$ can't be arbitrary. If $P$ is discrete then you end up with a giant coproduct, I think.2012-10-01

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