“Baptize first the children, and if they can speak for themselves let them do so. Otherwise, let their parents or other relatives speak for them” (The Apostolic Tradition 21:16 [A.D. 215]).
“Every soul that is born into flesh is soiled by the filth of wickedness and sin. . . . In the Church, baptism is given for the remission of sins, and, according to the usage of the Church, baptism is given even to infants. If there were nothing in infants which required the remission of sins and nothing in them pertinent to forgiveness, the grace of baptism would seem superfluous” (Homilies on Leviticus 8:3 [A.D. 248]).
“The Church received from the apostles the tradition of giving baptism even to infants. The apostles, to whom were committed the secrets of the divine sacraments, knew there are in everyone innate strains of [original] sin, which must be washed away through water and the Spirit” (Commentaries on Romans 5:9 [A.D. 248]).
Cyprian of Carthage
“As to what pertains to the case of infants: You [Fidus] said that they ought not to be baptized within the second or third day after their birth, that the old law of circumcision must be taken into consideration, and that you did not think that one should be baptized and sanctified within the eighth day after his birth. In our council it seemed to us far otherwise. No one agreed to the course which you thought should be taken. Rather, we all judge that the mercy and grace of God ought to be denied to no man born” (Letters 64:2 [A.D. 253]).
“If, in the case of the worst sinners and those who formerly sinned much against God, when afterwards they believe, the remission of their sins is granted and no one is held back from baptism and grace, how much more, then, should an infant not be held back, who, having but recently been born, has done no sin, except that, born of the flesh according to Adam, he has contracted the contagion of that old death from his first being born. For this very reason does he [an infant] approach more easily to receive the remission of sins: because the sins forgiven him are not his own but those of another” (ibid., 64:5).
Gregory of Nazianz
“Do you have an infant child? Allow sin no opportunity; rather, let the infant be sanctified from childhood. From his most tender age let him be consecrated by the Spirit. Do you fear the seal [of baptism] because of the weakness of nature? Oh, what a pusillanimous mother and of how little faith!” (Oration on Holy Baptism 40:7 [A.D. 388]).
“‘Well enough,’ some will say, ‘for those who ask for baptism, but what do you have to say about those who are still children, and aware neither of loss nor of grace? Shall we baptize them too?’ Certainly *, if there is any pressing danger. Better that they be sanctified unaware, than that they depart unsealed and uninitiated” (ibid., 40:28).
I don’t see it “anywhere” Specifically Baptism is for the remission of sin in relation to Adam? Or is there explicit documentation otherwise from pre 300?
[quote]children, and aware neither of [loss nor of grace]?= Adam and Eve
For this very reason does he [an infant] approach more easily to receive the remission of sins: because the sins forgiven him are not his own but those of another" (ibid., 64:5).
born, has done no sin, except that, born of the flesh according to Adam,
Every soul that is born into flesh is soiled by the filth of wickedness and sin. . . . In the Church, baptism is given for the remission of sins
There are primary and secondary intentions. The primary is stated above and per Creed. Its the “remission of Sin” from Adam. The rest are secondary intentions.