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This is probably quite an elementary question but I'm stuck on it for some reason.

Suppose I have a monoidal category $(\mathcal C, \otimes, I, \alpha, \lambda, \rho)$ and a morphism $f : X\rightarrow Y$ in $\mathcal C$.

Consider the composite $X \xrightarrow{\rho^{-1}} X\otimes I \xrightarrow{f\otimes \text{id}_I} Y\otimes I \xrightarrow{\rho} Y$ from $X$ to $Y$. Is it true that

$$\rho\circ(f\otimes \text{id}_I)\circ\rho^{-1} = f,$$

and if so, how does one prove it?

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    Use the fact that $\rho$ is a natural transformation. – 2017-01-31

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