Given a variety abelian $ A $ defined over an algebraically closed field of characteristic $ 0 $, Mumford define $ Pic^0(A)$= $L \in Pic (A) | T^*_x{L}L = L \ for \ all \ x \ in A$ , where $T_x$ is translation by x.

I wonder if this coincides with the usual definition: $ Pic ^ 0 ( A )$ is the connected component of identity in $ Pic (A) $?