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Logan 100% gets jealous of Mary Puppins.
Wade always showers her with compliments, kisses her little head and lets her climb on his lap. The merc will do everything and more for that little dog and Logan wants all that attention for himself.
At first, he can't really identify the feeling as jealousy, so he just ignores the feeling and then one day, it hits him. Wade has just started his usual shtick of coddling Dogpool and is letting her slobber all over his face and Logan realises that he has never wanted to be a dog more than right this moment.