This is question 8 from chapter 4 section 2 of MacLane's Category for the Working Mathematician:
If the category J is connected, prove for any $c \in C$ that $Lim \triangle c \cong c$ and $Colim \triangle c \cong c$, where $\triangle c: J \rightarrow C$ is the constant functor.
I'm not sure what the connectedness of J means to the mapping $\triangle c$, since any morphism in J is mapped to the identity $1_c$ in C. The lack of intuition is what's bothering me the most. MacLane does not offer much on connectedness beyond its definition. Can anyone explain to me what the general idea here is?
You can get intuition by a counterexample, as ZhenLin suggested, let $\mathcal J$ be the discrete category of $2$ points. Then we get $$\overset{\leftarrow}\lim\,\Delta c\ =\ c\times c\quad \text{ and }\quad \overset{\to}\lim\,\Delta c\ =\ c+c\,,$$ where $+$ denotes coproduct here.
(Already in $\mathcal{Set}$ we have neither $A\times A\cong A$ nor $A+A\cong A$ for sets $A$ with cardinalty $\ge2$.)
Generally, a cone $\gamma$ from $c$ to a diagram (i.e. functor) $\ D:\mathcal J\to \mathcal C\ $ is a collection of morphisms $\gamma_j:c\to Dj$ (with $j\in Ob\,\mathcal J$) that makes all arising triangles commute, i.e. making $\ Df\circ\gamma_j=\gamma_{j_1}\ $ for all arrows $f:j\to j_1$ in $\mathcal J$.
In case $\mathcal J$ is discrete, this latter condition is vacuous, so any collection of morphisms $(\gamma_j:c\to Dj)_{j\in Ob\mathcal J}$ is a cone from $c$ to $D$, so by [co-]limit of $D$ we get back exactly the notion of [co-]product of objects $(Dj)_{j\in Ob\mathcal C}$.
The limit cone of a diagram $D$ might be visualized as the 'rightmost' cone among all cones to $D$. (Precisely: each cone from $c$ to $D$ factors through uniquely the limit cone, so we get one-to-one correspondence between cones $c\leadsto D$ and morphisms $c\to\displaystyle\overset\leftarrow\lim D\ $ of $\,\mathcal C$).
So, back to the exercise, if $\mathcal J$ is connected, then what are the cones to the constant diagram $\Delta c$?