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I am following Pavel Etingof et al's book on tensor categories.

They give FinVect as an example of a rigid monoidal category, with evaluation map given by $\text {ev}_V(\epsilon\otimes v)=\epsilon(v)$ and coevaluation map $\text{coev}_V:\mathbb K \rightarrow V\otimes V^*$ "the usual embedding".

My question is embarassingly simple: what is this "usual embedding?" I was trying to see what it had to be using the definition of evaluation and coevaluation, but got nowhere.

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What Etingof et al call the "usual embedding" is given by linear extension of the map$$1_K\mapsto\sum_{i = 1}^{n}v_i\otimes v_i^* $$

where the $v_i$ are a basis of V and $v_i^*$ denoted the basis dual to the $v_i$'s.

This makes perfect sense if you want $ev \circ f\otimes id\circ coev = Trace(f)$ for any $f:V\longrightarrow V$.

Try $f=id$ and you'll find that $$ev\circ coev = dim(V)$$

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    $\begingroup$ To rephrase things slightly, if $V$ is finite dimensional, there is an isomorphism with $\hom(V,V)$ and $V\otimes V^{*}$, and the usual embedding is the inclusion of scalar maps into all linear maps. $\endgroup$
    – Aaron
    Dec 4, 2018 at 11:52

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