Page 61, below an addendum Yoneda Lemma, at the bottom of the page, last sentence:
The object function $r\mapsto D(r,-)$ and the arrow function $$(f:s\rightarrow r)\mapsto D(f,-):D(r,-)\rightarrow D(s,-)$$ for f an arrow of $D$ together define a full and faithful functor.
I tried to prove first that this is really functor. If we name this functor $Y$, we have to prove that $Y_{1_r}=1_{Y_r}$ and this is quite easy, but I have some problems with proving $Y_{g\circ f}= Y_g \circ Y_f$. If I understand well and if $f:A\rightarrow B, g:B\rightarrow C$ and T is one object from $D$, left side is following: $(Y_{g\circ f})_T(h:C \rightarrow T)=h\circ (g \circ f)$ How to prove that left side is the same. I have confusion because $Y_g:D(C,-)\rightarrow D(B,-)$ and $Y_f:(D(B,-)\rightarrow D(A,-)$, so $(Y_g)_T:Y_C(T) \rightarrow Y_B(T)$ and $(Y_f)_T:Y_B(T) \rightarrow Y_A(T)$. Hence I can't compose right side, i.e. $((Y_g)_T \circ (Y_f)_T)(h:C\rightarrow T)$ is not well defined.
Where are my mistakes?