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Suppose that $\mathcal{C}$ has finite products. Why is the unit $\eta_c:c\to c\times c$ of the adjunction $\Delta\dashv \times:\mathcal{C}\times\mathcal{C}\to\mathcal{C}$ the arrow $\langle 1_c,1_c\rangle$? In other words, why is $\pi_1\circ \eta_c=1_c=\pi_2\circ\eta_c$?

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The adjunction is saying that morphisms $$(f,g) : (c,c) \to (a,b)$$ in $\mathcal{C} \times \mathcal{C}$ correspond naturally with morphisms $$h : c \to a \times b$$ in $\mathcal{C}$. This correspondence is fairly obviously given by putting $h=\langle f,g \rangle$, where $\langle f,g \rangle$ is the unique morphism $c \to a \times b$ induced by $f : c \to a$ and $g : c \to b$ by the universal property of the product. (And vice versa.)

The $c$-component of the unit is therefore the morphism corresponding to $1_{(c,c)}$. But $1_{(c,c)} = (1_c, 1_c)$, meaning that $$\eta_c = \langle 1_c, 1_c \rangle : c \to c \times c$$ Likewise the counit is the morphism corresponding with $1_{a \times b} : a \times b \to a \times b$. But $1_{a \times b} = \langle \pi_1, \pi_2 \rangle$, and so $$\varepsilon_{(a,b)} = (\pi_1, \pi_2) : (a \times b, a \times b) \to (a,b)$$

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Hmm, thanks! I actually tried to find the explicit form of the adjunct, and, if $\phi$ is the bijection of the adjunction, and $f:c\to a$, $g:c\to b$, I found via naturality, that $$\phi(f,g)=(f\times g)\circ \eta_c$$ I struggle a bit to see why this is precisely $\langle f,g\rangle$. Am I doing something wrong? – Niels.Remb05 Apr 25 '14 at 5:38
@Niels.Remb05: Right, and now $$(f \times g) \circ \eta_c = (f \times g) \circ \langle 1_c, 1_c \rangle = \langle f \circ 1_c, g \circ 1_c \rangle = \langle f,g \rangle$$ Remember $f \times g = \langle f \circ \pi_1, g \circ \pi_2 \rangle$. – Clive Newstead Apr 25 '14 at 19:46
... but this uses the fact that $\eta_c=\langle 1_c,1_c\rangle$, which is what I want to prove, doesn't it?...I was trying to prove that form of the adjunct, in order to prove that $\eta_c=\langle 1_c,1_c\rangle$...Isn't this your argument? – Niels.Remb05 Apr 25 '14 at 19:52
You can check adjunctions in many ways. Either find a natural isomorphism, then your unit and counit are given by what the identities map to; or define what you want to be your unit and counit and prove that they satisfy the required conditions (i.e. either they induce the appropriate natural isomorphism, or they satisfy the trangle identities, or whatever). I can't tell which way round you're working here. – Clive Newstead Apr 25 '14 at 20:37
So either (a) check that the isomorphism $(\mathcal{C} \times \mathcal{C})((c,c),(a,b)) \cong \mathcal{C}(c,a \times b)$ given by $(f,g) \mapsto \langle f,g \rangle$ is natural in $c$ and in $(a,b)$ and find what $1_{(c,c)}$ maps to to get your unit $\eta_c$; or (b) define $\eta_c = \langle 1_c, 1_c \rangle$ and verify that the map $(f,g) \mapsto (f \times g) \circ \eta_c$ induces a natural isomorphism on hom sets. (Likewise for the counit.) – Clive Newstead Apr 25 '14 at 20:38

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