# Lift uniqueness for connected colimits of the projection functor $\Pi : c/C \to C$

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!

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.