# Forgetful functor from the slice category creates limits

I am reading "Category theory in context" and I'm having difficulties with proposition 3.3.8, top of page 92, pdf.

Theorem: The forgetful functor $$F : c/C \to C$$ strictly creates all limits that $$C$$ admits.

Proof (paraphrased): Defining a diagram over a slice category $$J \to c/C$$ is equivalent to defining a functor $$K : J \to C$$ and a cone $$\kappa : \Delta c \Rightarrow K$$. Suppose $$K$$ has a limit cone $$\lambda: \Delta l \Rightarrow K$$, then we'll show that this cone lifts to a cone over the slice category.

Since $$\lambda$$ is a limit cone in $$C$$, there's a unique factorisation of the cone $$\kappa : \Delta c \Rightarrow K$$ through $$\lambda$$, call it $$t : c \to l$$.

The book says that it's straightforward to verify that $$t$$ is a limit for the diagram in $$c/C$$.

Clearly $$t$$ is a summit of the cone. I can take the $$i$$-th leg to be $$\lambda_i$$, which is a morphism in the slice category from $$t$$ to $$\kappa_i$$.

It remains to show this cone is a limit cone. If I have a cone with a summit $$v : c \to m$$ and legs $$\mu_i : m \to K(i)$$, then I have to show there's a unique morphism in the slice category from $$v$$ to $$t$$ that commutes with the legs. I see only one possible candidate: because $$\lambda$$ is a limit in $$C$$, there's a morphism $$p : m \to l$$ in $$C$$ that commutes with the legs. But why is $$p$$ a morphism in the slice category, i.e. why does $$p \circ v = t$$?

The equality $$p\circ v=t$$ follows from the fact that the $$\lambda_i$$ for $$i\in J$$, being a limit cone, are jointly monomorphic (also said to be a monosource) and $$\lambda_i\circ p\circ v=\mu_i\circ v=\kappa_i=\lambda_i\circ t$$ for every $$i$$.
• Since I didn’t know the word monosource until now, let me add that another way to say it is that the legs $\lambda_i$ of the limit cone are jointly monomorphic Commented Dec 25, 2021 at 17:33
• Thank you, I haven't encountered this term before. I now see that the cone can be factored through the limit cone $\lambda_i$ via $p \circ v$ and via $t$ therefore by uniqueness they are equal. Commented Dec 25, 2021 at 17:49