Question

If we consider an oriented locally finite graphs $X$, and we suppose that $X$ is the projective limit of a familly of oriented locally finite graphs : $X=\lim\limits_{\overleftarrow{k\in\mathbb{N}}}X_{k}$, Is it true that $$\displaystyle H_{c}^{1}(X,\mathbb{C})=\lim\limits_{\overrightarrow{k\in \mathbb{N}}}H_{c}^{1}(X_{k},\mathbb{C})$$ where for an oriented graph $Y$, $H_{c}^{1}(Y,\mathbb{C})$ is the cohomology with compact support of $Y$. We know that if $Y$ is locally finite, the cohomolgy with compact support of $Y$ is the same of the singular cohomology of the geometric realization $|Y|$ of $Y$, e.q : $$\displaystyle H_{c}^{1}(Y,\mathbb{C})=H^{1}(|Y|,\mathbb{C})$$