In Categories for the Working Mathematician by Mac Lane he illustrated the fact that a functor with left adjoint preserves limits from the aspect on compositions of adjoints. (Page 119)
If $< F, G, \eta , \epsilon > : X \rightarrow A$ is a adjuntion and limits of $T : J \rightarrow A$ exists. Then $G^J$ preserves limits $\lim T$.
Compositing the left and up, and the right and down adjunction respectively. Since $F^J \Delta = \Delta F$ $G \lim \cong \lim G^J$ (left adjoint to a given functor is unique up to isomorphism).
After proving $\lim G^J(T) = G \lim T$ we need to prove that the $G \tau$ is limiting cone where $\tau$ is the limiting cone of $A^J$.
Then Mac Lane said
Put units and counits in the square diagram above, and recall that the limiting cone $\tau : a \rightarrow T$ is just the value of the counit of the adjunction $<\Delta, Lim, \dots> A \rightarrow A^J$ on the function $T$
What's this actually doing?