If each basepoint is a deformation retract of an open neighborhood, $\pi_1$ of a wedge of path-connected spaces is the free product of their fundamental groups.Suggest a slogan
Let \((X, x_0)\) and \((Y, y_0)\) be path-connected pointed spaces such that \(x_0\) and \(y_0\) are deformation retracts of open neighborhoods in \(X\) and \(Y\) respectively. Then the inclusions induce an isomorphism \[ \pi_1(X)\ast\pi_1(Y)\xrightarrow{\ \sim\ }\pi_1(X \vee Y) \] [@Hat02].