Cofiber of inverse limit

Question

Let $$ X\to\dots\to X_i\to \dots \to X_1 $$ be an inverse system of spaces with $X=\varprojlim X_i$ together with maps (commuting with each other) $Y\xrightarrow{f_i} X_i$ with (homotopy) cofibers $C_i$. I'm interested in (homotopy) cofiber $C$ of the limit map $Y\to X$. I doubt that $C=\varprojlim C_i$, but still, is it possible to describe (exact sequence will be great) $\pi_1 C$ in terms of $C_i$ solely, without referring to $X_i$ ?