fa.functional analysis - Convergence of operators to the identity on Banach spaces

Let $U_infty$ be a compact space, and let $U_r$ be an increasing family of compact subspaces whose closure is all of $U_infty$. That is, $U_r subseteq U_{r'}$ if $r le r'$ and $U_infty = overline{bigcup U_r}$.

For $r in [1,infty]$, let $Y_r = C(U_r,mathbb R)$ be the Banach space of real-valued continuous functions over $U_r$ with the supremum norm. For $r le r'$, let $phi_{r,r'} : Y_{r'} to Y_r$ be the restriction maps, so that $Y_infty$ is the inverse limit of the spaces $Y_r$. Write $phi_r : Y_infty to Y_r$ for the restriction map $phi_{r,infty}$.

Suppose there exists a family of continuous linear operators $m_r : Y_r to Y_infty$ such that $|m_r| le M$ for all $r$, and $phi_r circ m_r$ is the identity map on $Y_r$.

Question: Suppose $Gamma subseteq Y_infty$ is compact. Does $m_r circ phi_r$ converge strongly to the identity operator on $Gamma$? That is, for all $epsilon > 0$, does there exist $R > 0$ such that if $r ge R$, then $$sup_{y in Gamma} left| (m_r circ phi_r)(y) - y right|_{Y_infty} < epsilon?$$