On page 234 of Categories for the working mathematician, Mr. Saunders Mac Lane says followings:
Theorem 2. A functor $G:A\rightarrow X$ has a left adjoint if and only if both (1) $G$ preserves all limits which exists in $A$, and (2) for all objects $x$ of $X$, $Lim(Q:(x\downarrow G)\rightarrow A)$ exists in $A$, where $Q$ is the projection $Q\langle g, a\rangle=a$.
In a part of this proof, he says "Since a left adjoint $F$ to $G$ has each $\langle \eta_x :x\rightarrow GFx, Fx\rangle$ an initial object in $(x\downarrow G)$, any functor on this comma category has a limit (namely, its value on that initial object)."
I can understand that $\langle \eta_x :x\rightarrow GFx, Fx\rangle$ forms initial. But I can not do that it follows the result. How do I show it?
Say you have a category $C$ with an initial object $I$. Take a functor $F:C\rightarrow D$, then clearly $(F(I),\{h_X=F(f_!):F(I)\rightarrow F(X)\}_{X\in Ob(C)})$ is a cone for $F$ (here $f_!:I\rightarrow X$ is the unique morphism $I\rightarrow X$) and you can check this by simply checking the definition.
Moreover, if $(L,\{g_X:L\rightarrow F(X)\}_{X\in Ob(C)})$ is another cone, there is a unique map $\alpha:L\rightarrow F(I)$ such that $$g_X=h_X\circ \alpha$$This unique $\alpha$ is none other than $g_I$ by definition of cone (that is, for every $f:X\rightarrow Y$, $g_Y=F(f)\circ g_X$)