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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 $B?$

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up vote 2 down vote accepted

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.

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and $P$ should be small? –  Paul Slevin Sep 28 '12 at 14:16
    
Well, I guess so, if "cocomplete" means, as in Mac Lane's, "small cocomplete". –  a.r. Sep 28 '12 at 14:25
    
I am wondering though if it is true for any monoidal category which is cocomplete, even if the category $P$ is large. Does anyone know? –  Paul Slevin Sep 28 '12 at 16:24
    
I think this would be a question on its own. But without the monoidal thing messing around: just two categories $P$ and $C$ and what does it mean for $C$ to be "complete" or "cocomplete": $P$ small? Large? –  a.r. Sep 29 '12 at 17:14
    
a category has all limits means that it is a preorder (at most one morphism between any two objects). So we never consider it because it is pretty useless. So complete is said to mean has all small limits –  Paul Slevin Sep 29 '12 at 17:38
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