Lesson from Pius XII: Answer is “yes”. As man, with a human mind, Jesus knew each of us, and very intimately. As he hung on the Cross, he had us in mind. Hence, truly, he gave himself ‘for me’ as Paul proclaims. Let us honor the King of Kings.
For, as the Spaniards say, “He is fully man, yes, but no vulgar man.” Let us not measure his humanity against the narrow confines of our imagination.
From his marvelous encyclical Mystici corporis, art. 75:
75. Now the only-begotten Son of God embraced us in His infinite knowledge and undying love even before the world began. And that He might give a visible and exceedingly beautiful expression to this love, He assumed our nature in hypostatic union: hence – as Maximus of Turin with a certain unaffected simplicity remarks – “in Christ our own flesh loves us.” But the knowledge and love of our Divine Redeemer, of which we were the object from the first moment of His Incarnation, exceed all that the human intellect can hope to grasp. For hardly was He conceived in the womb of the Mother of God, when He began to enjoy the Beatific Vision, and in that vision all the members of His Mystical Body were continually and unceasingly present to Him, and He embraced them with His redeeming love. O marvelous condescension of divine love for us! O inestimable dispensation of boundless charity! In the crib, on the Cross, in the unending glory of the Father, Christ has all the members of the Church present before Him and united to Him in a much clearer and more loving manner than that of a mother who clasps her child to her breast, or than that with which a man knows and loves himself.
Pius is in fact simply reiterating, though authoritatively, the Tradition.