I have been having some problems with exercise 1.4.A in Vakil's Foundation of Algebraic Geometry. I feel like it's really basic (it's marked as a "reality check") and yet I can't seem to solve it.
1.4.A. Suppose that the partially ordered set $\mathcal{J}$ has an initial object $e$. Show that the limit of any diagram indexed by $\mathcal{J}$ exists.
Proof attempt. Since $e$ is initial in $\mathcal{J}$, for every $x \neq e$ of $\mathcal{J}$ there is a unique arrow $f_x : e \to x$. Given a diagram $F : \mathcal{J} \to \mathcal{C}$, my idea was to have its limit be $F(e)$ along with all the arrows $F(f_x)$. First we prove that it is a cone: if $f : x \to y$ is any arrow in $\mathcal{J}$ then the triangle
commutes because $$ F(f) \circ F(f_x) = F(f \circ f_x) = F(f_y), $$ where the last equality follows from the fact that $f \circ f_x$ and $f_y$ are both arrows $e \to y$ and are thus equal (equality can come either from $\mathcal{J}$ being a poset or from $e$ being initial). Hence this is a cone. To be a limit, it needs to satisfy the following universal property: given any object $H$ in $\mathcal{C}$ and arrows $g_x : H \to F(x)$ and $g_y : H \to F(y)$ with $F(f) \circ g_x = g_y$, there exists a unique arrow $g : H \to F(e)$ through which $g_x$ and $g_y$ factor, i.e. this diagram
commutes.
Here is where I get stuck. I can't find any "obvious" maps $H \to F(e)$, let alone any that make the diagram commute. I can't even make use of the fact that $e$ is initial, since the arrow goes in the wrong direction (it wouldn't matter anyway, since $H$ need not be in the image of $F$). Moreover, I haven't used the fact that $\mathcal{J}$ is a poset, since the commutativity of the first diagram could have just been proven using the fact that $e$ is initial.
So, my questions are:
Am I right in thinking that the limit is $F(e)$ (with the unique maps)?
If so, how can I prove the existence of $g$? What am I missing?
Does the poset structure matter or could the hypothesis be broadened to small categories $\mathcal{J}$ with an initial object?
I'm almost sure I am missing something tremendously trivial, so a super small hint should be fine. Thank you in advance for your help.
There's a map $H\to F(e)$ given by assumption, namely $g_e$!
As to your other questions, yes, you have the correct limit; no, it does not matter that $\mathcal J$ is a poset.