What is $\mathbf{B}\Omega A$, where $A$ is a pointed object of an $(\infty,1)$ category with point $*\to A$, $\Omega A$ is the loop space of $A$, and $\mathbf{B}X$ is the delooping of $X$?

The closest I have come to finding anything about this is in this $n$lab entry, titled *looping*, it is mentioned that the based loop space object of $A$, i.e., $\mathbf{B}\Omega_{\mathrm{pt}}A\simeq A$.

**Edit:** How does one prove the following statement: *there is a map $\mathbf B\Omega A \to A$ which is a weak equivalence onto the connected component of the base point*, which is given in an answer by John Klein below?