
nachsehen
to look up
To check something by searching for information in a particular place, for example in a book or on the internet.
Ich muss die genaue Adresse im Internet nachsehen.
I need to look up the exact address on the internet.
to look up / check / forgive
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to look up
To check something by searching for information in a particular place, for example in a book or on the internet.
Ich muss die genaue Adresse im Internet nachsehen.
I need to look up the exact address on the internet.

to check on
To go and check whether everything is all right.
Kannst du bitte mal nach den Kindern sehen, es ist so still im Zimmer.
Could you please go check on the children? It's so quiet in the room.

to forgive someone for something
To forgive someone for a mistake or misbehavior and not hold it against them.
Sieh es mir bitte nach, dass ich deinen Geburtstag vergessen habe.
Please forgive me for forgetting your birthday.

to watch someone go
To follow someone or something with your eyes as they move away.
Sie stand am Fenster und sah dem Zug nach, bis er am Horizont verschwand.
She stood at the window and watched the train go until it disappeared on the horizon.
'Nachsehen' is a separable verb. The prefix 'nach-' comes at the end of the main clause. In subordinate clauses and in the infinitive with 'zu', the verb stays together.
Ich sehe schnell die Öffnungszeiten nach.
I'll quickly look up the opening hours.
The case of the object helps you recognize the meaning. 'Etwas (Akkusativ) nachsehen' means to check information. 'Jemandem (Dativ) nachsehen' means to watch a person as they leave. 'Jemandem (Dativ) etwas (Akkusativ) nachsehen' means to forgive a mistake.
Kannst du die Telefonnummer nachsehen? (Information)
Can you look up the phone number? (information)
When it is about checking information, the place where you search is often mentioned. Typical phrases are 'im Wörterbuch nachsehen' (to look up in the dictionary), 'im Internet nachsehen' (to look up on the internet), or 'in den Unterlagen nachsehen' (to check in the documents).
Wenn du das Wort nicht kennst, sieh doch im Wörterbuch nach.
If you don't know the word, just look it up in the dictionary.
In the meaning 'to check whether everything is all right', the preposition 'nach' + dative is often used. Another common construction is a subordinate clause with 'ob' (whether/if).
Der Vater geht nachts oft nach seinem kranken Sohn sehen.
The father often goes to check on his sick son at night.
In the meaning 'to forgive', the verb always needs a person in the dative. This is the person whose mistake you forgive. The mistake itself is in the accusative.
Bitte sieh es mir nach, ich habe es nicht mit Absicht getan.
Please forgive me for it; I didn't do it on purpose.
to assign / to forgive / to miss
Both words can express that you no longer hold someone’s mistake against them. Vergeben suggests complete forgiveness, while jemandem etwas nachsehen often refers to tolerantly overlooking a minor mistake. For a serious offense, vergeben is more appropriate; for a small mishap, nachsehen is often the better choice.

Ich sehe dir den kleinen Fehler bei der Präsentation nach.
I forgive you for the small mistake you made during the presentation.
Nach seiner ehrlichen Entschuldigung konnte ich ihm den schweren Fehler vergeben.
After his sincere apology, I was able to forgive him for the serious mistake.