In Folland's A Course in Abstract Harmonic Analysis, he defines for a function $f$ on a topological group $G$, and $y\in G$,
$$L_{y}f:x\to f(y^{-1}x)\text{ and }R_{y}f:x\to f(xy)$$
He then remarks that the $y$ is inverted so that the map $y\mapsto L_{y}$ is a group homomorphism.
But it seems to me that $L_{yx}f = f((yx)^{-1}\cdot) = f(x^{-1}y^{-1}\cdot) = \left[L_{x}\circ L_{y}\right]f$, making the map an anti-homomorphism.
Am I misunderstanding the remark?
For any $z \in G$, $$ \begin{align} \left( L_{yx}f \right)(z) &= f((yx)^{-1}z) \\ &= f(x^{-1} y^{-1} z) \\ &= \left( L_x f \right)(y^{-1}z) \\ &= \left( L_y L_x f \right)(z). \end{align} $$ So, $$ L_{yx} f = L_y L_x f $$ for all $f \in G^*$.