For context, I am reading Weibel's k-book and I am trying to express the homology of $BS^{-1}S$, the group completion of the classifying space of a symmetric monoidal category, as a colimit. In particular, if we denote by $Y_S$ the base point component of $BS^{-1}S$ so that this space is homeomorphic to $\pi_0(BS^{-1}S)\times Y_S$, the result I want is
$$H_q(Y_S)\cong \varinjlim_{s\in\pi_0(S)\int\pi_0(S)}H_q(Aut(s))$$
Where we choose a specific representative of each element of $\pi_0(S)$ and $\pi_0(S)\int\pi_0(S)$ is the translation category of the monoid $\pi_0(S)$.
In doing so I arrive at the following
$$H_q(BS^{-1}S)\cong \varinjlim_{s\in\pi_0(S)\int\pi_0(S)}\bigsqcup_{s\in\pi_0(S)}H_q(Aut(s))$$
As both $\varinjlim$ and $\bigsqcup$ are colimits, I would like to naively exchange them to get
$$H_q(BS^{-1}S)\cong \bigsqcup_{s\in\pi_0(S)}\varinjlim_{s\in\pi_0(S)\int\pi_0(S)}H_q(Aut(s))$$
Except that the result I would like is rather
$$H_q(BS^{-1}S)\cong \bigsqcup_{s\in Gr(\pi_0(S))}\varinjlim_{s\in\pi_0(S)\int\pi_0(S)}H_q(Aut(s))$$
Where $Gr(\pi_0(S))$ is the group completion of $\pi_0(S)$ (or in our case $\pi_0(BS^{-1}S)$.
All of this blabbering to ask, is their some subtlety I am missing when exchanging the two colimits $\varinjlim_{s\in\pi_0(S)\int\pi_0(S)}$ and $\bigsqcup_{s\in\pi_0(S)}$