factorization of a unit of an adjunction

Question

Let $F$ be a left adjoint functor to $V$. Factor $X \to VK$ through the adjunction unit $$ X \to VFX \to VK, $$ where the first map is $\eta_X$, the second map is $V$ of the adjoint map $FX \to K$, the composite is the given $X \to VK$. Why?

Answer 1

This follows from naturality of $\eta$ and one of the triangle identities: the adjoint of $f : X \to VK$ is $\bar f := \varepsilon_K \circ F(f) : FX \to K$, and so you have $$V(\bar f) \circ \eta_X = V(\varepsilon_K) \circ VF(f) \circ \eta_X \overset{\text{naturality}}{=} V(\varepsilon_K) \circ \eta_{VK} \circ f \overset{\text{triangle}}{=} f$$