I'm currently working through proposition 3.3.8 of Emily Riehl's Category Theory in Context, which proves that the projection functor $\Pi: c/C \to C$ strictly creates limits and connected colimits (i.e. if the image of a diagram is a (co) limit in $C$, there is a unique lift to a (co) limit on $c/C$). The first remark notes that a diagram $(K,\kappa) : J \to c/C$ in $c/C$ is a functor $K: J \to C$ together with a cone $\kappa: c \Rightarrow K$, whose image via $\Pi$ is the diagram $K$. Hence the idea is to prove that if $K$ is a limit/connected colimit then there is a unique (co) limit cone over $(K,\kappa)$ whose image is the (co) limit cone over $K$.
I have understood the case for limits, but there is a subtlety in the connected colimits case which I am failing to understand. The author takes a colimit cone $\mu : K \Rightarrow p$, and defines an arrow $c \xrightarrow{\zeta} p \in \operatorname{obj} c/C$ via $\zeta := \mu_j\kappa_j$ for some $j \in \operatorname{obj} J$. Immediately after, it is claimed that $\zeta$ is independent of the choice of $j$ since $J$ is assumed to be connected. Hence $\mu$ together with $\zeta$ give a colimit cone over $(K,\kappa)$, proving that $K$ has a colimit lift, and moreover it is unique since $\zeta$ is determined by $\mu$ and $\kappa$.
I get the outline of the argument, but I am not yet convinced of why $J$ being connected implies that $\zeta = \mu_j\kappa_j$ for all $j$ objects of $J$, which seems a central step in the proof (both for uniqueness and to define a lift cone in the slice category to begin with).
Any help would be greatly appreciated!
Best Answer
Pick any projection of the cone $\kappa$, i.e. $\kappa_j : c \to Kj$. Then for any other projection $\kappa_i : c \to Ki$, we have either $\kappa_j=Kf\circ\kappa_i$ or $\kappa_i=Kg\circ\kappa_j$ by connectedness. For the colimiting cocone, we have the opposite: given the coprojection $\mu_j : Kj\to p$ we have $\mu_i=\mu_j\circ Kf$ or $\mu_j=\mu_i\circ Kg$ respectively. In the first case, we have $\mu_j\circ\kappa_j = \mu_i\circ Kf\circ\kappa_i = \mu_i\circ\kappa_i$ and similarly for the second case.