In studying deformation theory of Galois representations, I've come surely to an error, relating Schlessinger's criterion.

Let's fix a representation $\bar{\rho}$ of a group $G$ and let $D_{\bar{\rho}}$ be its deformation functor from the category $\hat{\mathcal{C}}$ of complete noetherian local $W(\mathbb{F})$-algebras (where $W(\mathbb{F})$ is the ring of the Witt vectors over a finite field $\mathbb{F}$) with residue field $\mathbb{F}$. Let $A'$, $A''$ and $A$ be artinian objects of $\hat{\mathcal{C}}$, and let be given two homomorphisms of algebras $u' : A' \rightarrow A$ and $u'': A'' \rightarrow A$. One of the conditions of Schlessinger's criterion, namely that the natural map $$D_{\bar{\rho}}(A'\times_A A'') \rightarrow D_{\bar{\rho}}(A') \times_{D_{\bar{\rho}}(A)} D_{\bar{\rho}}(A'')$$ should be surjective when $u'$ is a small surjection, seems to me always true, regardless the surjectivity of $u'$. Take in fact $([\rho'], [\rho'']) \in D_{\bar{\rho}}(A') \times_{D_{\bar{\rho}}(A)} D_{\bar{\rho}}(A'')$ and let $\rho: G \rightarrow \operatorname{GL}_n(A'\times_A A'')$ be defined as $\rho(\sigma) = (\rho'(\sigma), \rho''(\sigma))$, using the identification $$\operatorname{GL}_n(A'\times_A A'') \simeq \operatorname{GL}_n(A')\times_{\operatorname{GL}_n(A)}\operatorname{GL}_n(A'').$$ By the property of this last identification, we see that by conjugating one of $\rho'$ or $\rho''$, we conjugate also $\rho$. Thus this yields a well defined $[\rho] \in D_{\bar{\rho}}(A'\times_A A'')$ and seems just the one we wanted.

Since this must clearly be wrong, where is my mistake?

Sorry if this question reveals my big miscomprehension! Thank you in advance.