I actually don't think that's correct Berze, they agree when there is a particle or when it's with essere not with avere. Ho mangiato, l'ho mangiato - the la/lo is the l'ho it. I've never heard it agree like l'ho mangiata.

Berze, thanks so much!

You have implied what you ate (object) in your sentence. You could have said "Si ho mangiato una pizza" but you took the object out and used the "la" pronoun and the verb goes with the gender of la/lo/le/li, Correct usage of spesa (as in grocery run) is "Avete fatto la spesa" and you can say "Si l'abbiamo fatta" (implicit) or "Si ho fatto la spesa" (explicit). For a general shopping run, you can say "Avete fatto spese? Si, le abbiamo fatte" or (explicit) "Si abbiamo fatto spese". "When accusative pronouns are used in a compound tense, the final vowel of the past participle must agree in gender and number with the accusative pronoun. For example, hai comprato i cocomeri e le mele? ("Did you buy the watermelons and the apples?") - Li [i cocomeri] ho comprati ma non le [le mele] ho comprate ("I bought them [the former] but I did not buy them [the latter]"). This also happens when the underlying pronoun is made opaque by elision: l'ho svegliato ("I woke him up"), versus L'ho svegliata ("I woke her up")." https://en.wikipedia.org/wiki/Italian_grammar#Pronouns

So it's like a "control" so people know what noun you are talking about? So... La ho manigato. L'ho mangiata. What about... Avete comprato le spese? Si'...l'abbiamo comprate. Is that correct?