This is a crosspost of this question from MSE.

I'm confused about the definition of a projective family of probability spaces $(S_t,\mathscr S _t,\mu_t,f_{ts})_{s,t\in T}$. The conditions

• $f_{tt}=1_{S_t}$
• $f_{us}=f_{ts}\circ f_{ut}$ where $s\leq t\leq u$

are clear from the usual definition of a projective system as a functor from $\mathsf T^\text{op}$, the opposite poset category. However, (in addition to measurability) the following condition always appears:

• $\mu_s = (f_{ts})_{\ast}(\mu_t)$, where the RHS is the pushforward measure.

This looks to me like a kind of cone coherence condition, but it does not (as far as I can see) fall out of the categorical formalism of limits if one works in the category with probability spaces as objects and measurable maps as arrows.

Furthermore, the projective limit (if it exists) is required to satisfy the following two conditions:

• Its $\sigma$-algebra is generated by the restrictions of the projections $\pi_t$
• Its probability measure $\mu$ must satisfy $(\pi_t)_\ast (\mu)=\mu _t$

In what category should one work so that all these conditions fall out of the categorical formalism? In other words, in what category does the categorical notion of projective limit coincide exactly with the one found in books.