# Adjoint question in the Category

If $\langle F,G,φ \rangle : X \to A$ is adjunction with $G$ full and every unit $η_x$ a monic, then every $η_x$ is also epi.

Some similar questions like this, maybe I do not catch the key, who can help me?

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The first sentence of your question is quoted verbatim from exercise 3 on p.92 of Mac Lane's Categories for the working mathematician. Are you asking how to solve this exercise? – t.b. May 16 '11 at 10:49
In Mac Lane's book,it is <G,F,Φ>，maybe that is a misprinting. – Strongart May 18 '11 at 10:27

Let $f, g \colon GF(A) \rightarrow B$ be two parallel morphisms and assume that $f \circ \eta_A = g \circ \eta_A = r$. We have to show that $f = g$.
Postcompose $r$ with $\eta_B$ to obtain a morphism $A \rightarrow GF(B)$. By universality of $\eta_A$, we know that there exists exactly one morphism $h \colon F(A) \rightarrow F(B)$ such that $G(h) \circ \eta_A = \eta_B \circ r$. But $G$ is full, so every morphism $GF(A) \rightarrow GF(B)$ is of the form of $G(k)$ and thus there exists exactly one morphism $GF(A) \rightarrow GF(B)$ such that the above condition is satisfied. Obviously, both $\eta_B \circ f$ and $\eta_B \circ g$ make the triangle commute. So, $\eta_B \circ f = \eta_B \circ g$, what gives $f = g$ beacuse $\eta_B$ is mono.